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Properties

Label 1785.2.a.g.1.1
Level $1785$
Weight $2$
Character 1785.1
Self dual yes
Analytic conductor $14.253$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1785,2,Mod(1,1785)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1785, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1785.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1785 = 3 \cdot 5 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1785.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(14.2532967608\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1785.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -2.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -2.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} +2.00000 q^{12} +3.00000 q^{13} -1.00000 q^{15} +4.00000 q^{16} +1.00000 q^{17} -2.00000 q^{19} -2.00000 q^{20} +1.00000 q^{21} -3.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{28} +4.00000 q^{29} -5.00000 q^{31} +2.00000 q^{33} -1.00000 q^{35} -2.00000 q^{36} -1.00000 q^{37} -3.00000 q^{39} +11.0000 q^{41} +4.00000 q^{43} +4.00000 q^{44} +1.00000 q^{45} -9.00000 q^{47} -4.00000 q^{48} +1.00000 q^{49} -1.00000 q^{51} -6.00000 q^{52} -2.00000 q^{55} +2.00000 q^{57} -12.0000 q^{59} +2.00000 q^{60} -5.00000 q^{61} -1.00000 q^{63} -8.00000 q^{64} +3.00000 q^{65} -12.0000 q^{67} -2.00000 q^{68} +3.00000 q^{69} -2.00000 q^{71} -4.00000 q^{73} -1.00000 q^{75} +4.00000 q^{76} +2.00000 q^{77} -10.0000 q^{79} +4.00000 q^{80} +1.00000 q^{81} -5.00000 q^{83} -2.00000 q^{84} +1.00000 q^{85} -4.00000 q^{87} -14.0000 q^{89} -3.00000 q^{91} +6.00000 q^{92} +5.00000 q^{93} -2.00000 q^{95} -2.00000 q^{97} -2.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) −1.00000 −0.577350
\(4\) −2.00000 −1.00000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 2.00000 0.577350
\(13\) 3.00000 0.832050 0.416025 − 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 4.00000 1.00000
\(17\) 1.00000 0.242536
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) −2.00000 −0.447214
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −3.00000 −0.625543 −0.312772 − 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 2.00000 0.377964
\(29\) 4.00000 0.742781 0.371391 − 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) −5.00000 −0.898027 −0.449013 − 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) 0 0
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) −2.00000 −0.333333
\(37\) −1.00000 −0.164399 −0.0821995 − 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) 0 0
\(39\) −3.00000 −0.480384
\(40\) 0 0
\(41\) 11.0000 1.71791 0.858956 − 0.512050i \(-0.171114\pi\)
0.858956 + 0.512050i \(0.171114\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 4.00000 0.603023
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −9.00000 −1.31278 −0.656392 − 0.754420i \(-0.727918\pi\)
−0.656392 + 0.754420i \(0.727918\pi\)
\(48\) −4.00000 −0.577350
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −1.00000 −0.140028
\(52\) −6.00000 −0.832050
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 0 0
\(59\) −12.0000 −1.56227 −0.781133 − 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 2.00000 0.258199
\(61\) −5.00000 −0.640184 −0.320092 − 0.947386i \(-0.603714\pi\)
−0.320092 + 0.947386i \(0.603714\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) −8.00000 −1.00000
\(65\) 3.00000 0.372104
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 − 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) −2.00000 −0.242536
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 − 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 4.00000 0.458831
\(77\) 2.00000 0.227921
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 − 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 4.00000 0.447214
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −5.00000 −0.548821 −0.274411 − 0.961613i \(-0.588483\pi\)
−0.274411 + 0.961613i \(0.588483\pi\)
\(84\) −2.00000 −0.218218
\(85\) 1.00000 0.108465
\(86\) 0 0
\(87\) −4.00000 −0.428845
\(88\) 0 0
\(89\) −14.0000 −1.48400 −0.741999 − 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) −3.00000 −0.314485
\(92\) 6.00000 0.625543
\(93\) 5.00000 0.518476
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) −2.00000 −0.200000
\(101\) −8.00000 −0.796030 −0.398015 − 0.917379i \(-0.630301\pi\)
−0.398015 + 0.917379i \(0.630301\pi\)
\(102\) 0 0
\(103\) 16.0000 1.57653 0.788263 − 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 0 0
\(105\) 1.00000 0.0975900
\(106\) 0 0
\(107\) 13.0000 1.25676 0.628379 − 0.777908i \(-0.283719\pi\)
0.628379 + 0.777908i \(0.283719\pi\)
\(108\) 2.00000 0.192450
\(109\) −10.0000 −0.957826 −0.478913 − 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 1.00000 0.0949158
\(112\) −4.00000 −0.377964
\(113\) −6.00000 −0.564433 −0.282216 − 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −3.00000 −0.279751
\(116\) −8.00000 −0.742781
\(117\) 3.00000 0.277350
\(118\) 0 0
\(119\) −1.00000 −0.0916698
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) −11.0000 −0.991837
\(124\) 10.0000 0.898027
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 − 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) −3.00000 −0.262111 −0.131056 − 0.991375i \(-0.541837\pi\)
−0.131056 + 0.991375i \(0.541837\pi\)
\(132\) −4.00000 −0.348155
\(133\) 2.00000 0.173422
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −8.00000 −0.683486 −0.341743 − 0.939793i \(-0.611017\pi\)
−0.341743 + 0.939793i \(0.611017\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 − 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 2.00000 0.169031
\(141\) 9.00000 0.757937
\(142\) 0 0
\(143\) −6.00000 −0.501745
\(144\) 4.00000 0.333333
\(145\) 4.00000 0.332182
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 2.00000 0.164399
\(149\) −11.0000 −0.901155 −0.450578 − 0.892737i \(-0.648782\pi\)
−0.450578 + 0.892737i \(0.648782\pi\)
\(150\) 0 0
\(151\) 19.0000 1.54620 0.773099 − 0.634285i \(-0.218706\pi\)
0.773099 + 0.634285i \(0.218706\pi\)
\(152\) 0 0
\(153\) 1.00000 0.0808452
\(154\) 0 0
\(155\) −5.00000 −0.401610
\(156\) 6.00000 0.480384
\(157\) −10.0000 −0.798087 −0.399043 − 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3.00000 0.236433
\(162\) 0 0
\(163\) 19.0000 1.48819 0.744097 − 0.668071i \(-0.232880\pi\)
0.744097 + 0.668071i \(0.232880\pi\)
\(164\) −22.0000 −1.71791
\(165\) 2.00000 0.155700
\(166\) 0 0
\(167\) 2.00000 0.154765 0.0773823 − 0.997001i \(-0.475344\pi\)
0.0773823 + 0.997001i \(0.475344\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) −8.00000 −0.609994
\(173\) −6.00000 −0.456172 −0.228086 − 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) −8.00000 −0.603023
\(177\) 12.0000 0.901975
\(178\) 0 0
\(179\) −19.0000 −1.42013 −0.710063 − 0.704138i \(-0.751334\pi\)
−0.710063 + 0.704138i \(0.751334\pi\)
\(180\) −2.00000 −0.149071
\(181\) −1.00000 −0.0743294 −0.0371647 − 0.999309i \(-0.511833\pi\)
−0.0371647 + 0.999309i \(0.511833\pi\)
\(182\) 0 0
\(183\) 5.00000 0.369611
\(184\) 0 0
\(185\) −1.00000 −0.0735215
\(186\) 0 0
\(187\) −2.00000 −0.146254
\(188\) 18.0000 1.31278
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −27.0000 −1.95365 −0.976826 − 0.214036i \(-0.931339\pi\)
−0.976826 + 0.214036i \(0.931339\pi\)
\(192\) 8.00000 0.577350
\(193\) 23.0000 1.65558 0.827788 − 0.561041i \(-0.189599\pi\)
0.827788 + 0.561041i \(0.189599\pi\)
\(194\) 0 0
\(195\) −3.00000 −0.214834
\(196\) −2.00000 −0.142857
\(197\) −18.0000 −1.28245 −0.641223 − 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −12.0000 −0.850657 −0.425329 − 0.905039i \(-0.639842\pi\)
−0.425329 + 0.905039i \(0.639842\pi\)
\(200\) 0 0
\(201\) 12.0000 0.846415
\(202\) 0 0
\(203\) −4.00000 −0.280745
\(204\) 2.00000 0.140028
\(205\) 11.0000 0.768273
\(206\) 0 0
\(207\) −3.00000 −0.208514
\(208\) 12.0000 0.832050
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −14.0000 −0.963800 −0.481900 − 0.876226i \(-0.660053\pi\)
−0.481900 + 0.876226i \(0.660053\pi\)
\(212\) 0 0
\(213\) 2.00000 0.137038
\(214\) 0 0
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) 5.00000 0.339422
\(218\) 0 0
\(219\) 4.00000 0.270295
\(220\) 4.00000 0.269680
\(221\) 3.00000 0.201802
\(222\) 0 0
\(223\) −15.0000 −1.00447 −0.502237 − 0.864730i \(-0.667490\pi\)
−0.502237 + 0.864730i \(0.667490\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 2.00000 0.132745 0.0663723 − 0.997795i \(-0.478857\pi\)
0.0663723 + 0.997795i \(0.478857\pi\)
\(228\) −4.00000 −0.264906
\(229\) 14.0000 0.925146 0.462573 − 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) −2.00000 −0.131590
\(232\) 0 0
\(233\) 3.00000 0.196537 0.0982683 − 0.995160i \(-0.468670\pi\)
0.0982683 + 0.995160i \(0.468670\pi\)
\(234\) 0 0
\(235\) −9.00000 −0.587095
\(236\) 24.0000 1.56227
\(237\) 10.0000 0.649570
\(238\) 0 0
\(239\) −15.0000 −0.970269 −0.485135 − 0.874439i \(-0.661229\pi\)
−0.485135 + 0.874439i \(0.661229\pi\)
\(240\) −4.00000 −0.258199
\(241\) 17.0000 1.09507 0.547533 − 0.836784i \(-0.315567\pi\)
0.547533 + 0.836784i \(0.315567\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 10.0000 0.640184
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) −6.00000 −0.381771
\(248\) 0 0
\(249\) 5.00000 0.316862
\(250\) 0 0
\(251\) 16.0000 1.00991 0.504956 − 0.863145i \(-0.331509\pi\)
0.504956 + 0.863145i \(0.331509\pi\)
\(252\) 2.00000 0.125988
\(253\) 6.00000 0.377217
\(254\) 0 0
\(255\) −1.00000 −0.0626224
\(256\) 16.0000 1.00000
\(257\) −7.00000 −0.436648 −0.218324 − 0.975876i \(-0.570059\pi\)
−0.218324 + 0.975876i \(0.570059\pi\)
\(258\) 0 0
\(259\) 1.00000 0.0621370
\(260\) −6.00000 −0.372104
\(261\) 4.00000 0.247594
\(262\) 0 0
\(263\) 24.0000 1.47990 0.739952 − 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 14.0000 0.856786
\(268\) 24.0000 1.46603
\(269\) −21.0000 −1.28039 −0.640196 − 0.768211i \(-0.721147\pi\)
−0.640196 + 0.768211i \(0.721147\pi\)
\(270\) 0 0
\(271\) 26.0000 1.57939 0.789694 − 0.613501i \(-0.210239\pi\)
0.789694 + 0.613501i \(0.210239\pi\)
\(272\) 4.00000 0.242536
\(273\) 3.00000 0.181568
\(274\) 0 0
\(275\) −2.00000 −0.120605
\(276\) −6.00000 −0.361158
\(277\) 7.00000 0.420589 0.210295 − 0.977638i \(-0.432558\pi\)
0.210295 + 0.977638i \(0.432558\pi\)
\(278\) 0 0
\(279\) −5.00000 −0.299342
\(280\) 0 0
\(281\) −1.00000 −0.0596550 −0.0298275 − 0.999555i \(-0.509496\pi\)
−0.0298275 + 0.999555i \(0.509496\pi\)
\(282\) 0 0
\(283\) 28.0000 1.66443 0.832214 − 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) 4.00000 0.237356
\(285\) 2.00000 0.118470
\(286\) 0 0
\(287\) −11.0000 −0.649309
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) 8.00000 0.468165
\(293\) 14.0000 0.817889 0.408944 − 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 0 0
\(295\) −12.0000 −0.698667
\(296\) 0 0
\(297\) 2.00000 0.116052
\(298\) 0 0
\(299\) −9.00000 −0.520483
\(300\) 2.00000 0.115470
\(301\) −4.00000 −0.230556
\(302\) 0 0
\(303\) 8.00000 0.459588
\(304\) −8.00000 −0.458831
\(305\) −5.00000 −0.286299
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 − 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) −4.00000 −0.227921
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) −5.00000 −0.283524 −0.141762 − 0.989901i \(-0.545277\pi\)
−0.141762 + 0.989901i \(0.545277\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 − 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 0 0
\(315\) −1.00000 −0.0563436
\(316\) 20.0000 1.12509
\(317\) −5.00000 −0.280828 −0.140414 − 0.990093i \(-0.544843\pi\)
−0.140414 + 0.990093i \(0.544843\pi\)
\(318\) 0 0
\(319\) −8.00000 −0.447914
\(320\) −8.00000 −0.447214
\(321\) −13.0000 −0.725589
\(322\) 0 0
\(323\) −2.00000 −0.111283
\(324\) −2.00000 −0.111111
\(325\) 3.00000 0.166410
\(326\) 0 0
\(327\) 10.0000 0.553001
\(328\) 0 0
\(329\) 9.00000 0.496186
\(330\) 0 0
\(331\) −17.0000 −0.934405 −0.467202 − 0.884150i \(-0.654738\pi\)
−0.467202 + 0.884150i \(0.654738\pi\)
\(332\) 10.0000 0.548821
\(333\) −1.00000 −0.0547997
\(334\) 0 0
\(335\) −12.0000 −0.655630
\(336\) 4.00000 0.218218
\(337\) 2.00000 0.108947 0.0544735 − 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) −2.00000 −0.108465
\(341\) 10.0000 0.541530
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 3.00000 0.161515
\(346\) 0 0
\(347\) −12.0000 −0.644194 −0.322097 − 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 8.00000 0.428845
\(349\) 12.0000 0.642345 0.321173 − 0.947021i \(-0.395923\pi\)
0.321173 + 0.947021i \(0.395923\pi\)
\(350\) 0 0
\(351\) −3.00000 −0.160128
\(352\) 0 0
\(353\) −3.00000 −0.159674 −0.0798369 − 0.996808i \(-0.525440\pi\)
−0.0798369 + 0.996808i \(0.525440\pi\)
\(354\) 0 0
\(355\) −2.00000 −0.106149
\(356\) 28.0000 1.48400
\(357\) 1.00000 0.0529256
\(358\) 0 0
\(359\) 3.00000 0.158334 0.0791670 − 0.996861i \(-0.474774\pi\)
0.0791670 + 0.996861i \(0.474774\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 7.00000 0.367405
\(364\) 6.00000 0.314485
\(365\) −4.00000 −0.209370
\(366\) 0 0
\(367\) 26.0000 1.35719 0.678594 − 0.734513i \(-0.262589\pi\)
0.678594 + 0.734513i \(0.262589\pi\)
\(368\) −12.0000 −0.625543
\(369\) 11.0000 0.572637
\(370\) 0 0
\(371\) 0 0
\(372\) −10.0000 −0.518476
\(373\) 20.0000 1.03556 0.517780 − 0.855514i \(-0.326758\pi\)
0.517780 + 0.855514i \(0.326758\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 − 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 4.00000 0.205196
\(381\) 8.00000 0.409852
\(382\) 0 0
\(383\) 4.00000 0.204390 0.102195 − 0.994764i \(-0.467413\pi\)
0.102195 + 0.994764i \(0.467413\pi\)
\(384\) 0 0
\(385\) 2.00000 0.101929
\(386\) 0 0
\(387\) 4.00000 0.203331
\(388\) 4.00000 0.203069
\(389\) 11.0000 0.557722 0.278861 − 0.960331i \(-0.410043\pi\)
0.278861 + 0.960331i \(0.410043\pi\)
\(390\) 0 0
\(391\) −3.00000 −0.151717
\(392\) 0 0
\(393\) 3.00000 0.151330
\(394\) 0 0
\(395\) −10.0000 −0.503155
\(396\) 4.00000 0.201008
\(397\) 10.0000 0.501886 0.250943 − 0.968002i \(-0.419259\pi\)
0.250943 + 0.968002i \(0.419259\pi\)
\(398\) 0 0
\(399\) −2.00000 −0.100125
\(400\) 4.00000 0.200000
\(401\) −10.0000 −0.499376 −0.249688 − 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) 0 0
\(403\) −15.0000 −0.747203
\(404\) 16.0000 0.796030
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 2.00000 0.0991363
\(408\) 0 0
\(409\) 32.0000 1.58230 0.791149 − 0.611623i \(-0.209483\pi\)
0.791149 + 0.611623i \(0.209483\pi\)
\(410\) 0 0
\(411\) 8.00000 0.394611
\(412\) −32.0000 −1.57653
\(413\) 12.0000 0.590481
\(414\) 0 0
\(415\) −5.00000 −0.245440
\(416\) 0 0
\(417\) −12.0000 −0.587643
\(418\) 0 0
\(419\) 7.00000 0.341972 0.170986 − 0.985273i \(-0.445305\pi\)
0.170986 + 0.985273i \(0.445305\pi\)
\(420\) −2.00000 −0.0975900
\(421\) −21.0000 −1.02348 −0.511739 − 0.859141i \(-0.670998\pi\)
−0.511739 + 0.859141i \(0.670998\pi\)
\(422\) 0 0
\(423\) −9.00000 −0.437595
\(424\) 0 0
\(425\) 1.00000 0.0485071
\(426\) 0 0
\(427\) 5.00000 0.241967
\(428\) −26.0000 −1.25676
\(429\) 6.00000 0.289683
\(430\) 0 0
\(431\) 28.0000 1.34871 0.674356 − 0.738406i \(-0.264421\pi\)
0.674356 + 0.738406i \(0.264421\pi\)
\(432\) −4.00000 −0.192450
\(433\) 1.00000 0.0480569 0.0240285 − 0.999711i \(-0.492351\pi\)
0.0240285 + 0.999711i \(0.492351\pi\)
\(434\) 0 0
\(435\) −4.00000 −0.191785
\(436\) 20.0000 0.957826
\(437\) 6.00000 0.287019
\(438\) 0 0
\(439\) −24.0000 −1.14546 −0.572729 − 0.819745i \(-0.694115\pi\)
−0.572729 + 0.819745i \(0.694115\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −24.0000 −1.14027 −0.570137 − 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) −2.00000 −0.0949158
\(445\) −14.0000 −0.663664
\(446\) 0 0
\(447\) 11.0000 0.520282
\(448\) 8.00000 0.377964
\(449\) −6.00000 −0.283158 −0.141579 − 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) −22.0000 −1.03594
\(452\) 12.0000 0.564433
\(453\) −19.0000 −0.892698
\(454\) 0 0
\(455\) −3.00000 −0.140642
\(456\) 0 0
\(457\) 8.00000 0.374224 0.187112 − 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) 0 0
\(459\) −1.00000 −0.0466760
\(460\) 6.00000 0.279751
\(461\) −34.0000 −1.58354 −0.791769 − 0.610821i \(-0.790840\pi\)
−0.791769 + 0.610821i \(0.790840\pi\)
\(462\) 0 0
\(463\) −2.00000 −0.0929479 −0.0464739 − 0.998920i \(-0.514798\pi\)
−0.0464739 + 0.998920i \(0.514798\pi\)
\(464\) 16.0000 0.742781
\(465\) 5.00000 0.231869
\(466\) 0 0
\(467\) 33.0000 1.52706 0.763529 − 0.645774i \(-0.223465\pi\)
0.763529 + 0.645774i \(0.223465\pi\)
\(468\) −6.00000 −0.277350
\(469\) 12.0000 0.554109
\(470\) 0 0
\(471\) 10.0000 0.460776
\(472\) 0 0
\(473\) −8.00000 −0.367840
\(474\) 0 0
\(475\) −2.00000 −0.0917663
\(476\) 2.00000 0.0916698
\(477\) 0 0
\(478\) 0 0
\(479\) −11.0000 −0.502603 −0.251301 − 0.967909i \(-0.580859\pi\)
−0.251301 + 0.967909i \(0.580859\pi\)
\(480\) 0 0
\(481\) −3.00000 −0.136788
\(482\) 0 0
\(483\) −3.00000 −0.136505
\(484\) 14.0000 0.636364
\(485\) −2.00000 −0.0908153
\(486\) 0 0
\(487\) −23.0000 −1.04223 −0.521115 − 0.853487i \(-0.674484\pi\)
−0.521115 + 0.853487i \(0.674484\pi\)
\(488\) 0 0
\(489\) −19.0000 −0.859210
\(490\) 0 0
\(491\) 9.00000 0.406164 0.203082 − 0.979162i \(-0.434904\pi\)
0.203082 + 0.979162i \(0.434904\pi\)
\(492\) 22.0000 0.991837
\(493\) 4.00000 0.180151
\(494\) 0 0
\(495\) −2.00000 −0.0898933
\(496\) −20.0000 −0.898027
\(497\) 2.00000 0.0897123
\(498\) 0 0
\(499\) 16.0000 0.716258 0.358129 − 0.933672i \(-0.383415\pi\)
0.358129 + 0.933672i \(0.383415\pi\)
\(500\) −2.00000 −0.0894427
\(501\) −2.00000 −0.0893534
\(502\) 0 0
\(503\) 16.0000 0.713405 0.356702 − 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) 0 0
\(505\) −8.00000 −0.355995
\(506\) 0 0
\(507\) 4.00000 0.177646
\(508\) 16.0000 0.709885
\(509\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) 0 0
\(513\) 2.00000 0.0883022
\(514\) 0 0
\(515\) 16.0000 0.705044
\(516\) 8.00000 0.352180
\(517\) 18.0000 0.791639
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −42.0000 −1.84005 −0.920027 − 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) 0 0
\(523\) −35.0000 −1.53044 −0.765222 − 0.643767i \(-0.777371\pi\)
−0.765222 + 0.643767i \(0.777371\pi\)
\(524\) 6.00000 0.262111
\(525\) 1.00000 0.0436436
\(526\) 0 0
\(527\) −5.00000 −0.217803
\(528\) 8.00000 0.348155
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) −4.00000 −0.173422
\(533\) 33.0000 1.42939
\(534\) 0 0
\(535\) 13.0000 0.562039
\(536\) 0 0
\(537\) 19.0000 0.819911
\(538\) 0 0
\(539\) −2.00000 −0.0861461
\(540\) 2.00000 0.0860663
\(541\) 16.0000 0.687894 0.343947 − 0.938989i \(-0.388236\pi\)
0.343947 + 0.938989i \(0.388236\pi\)
\(542\) 0 0
\(543\) 1.00000 0.0429141
\(544\) 0 0
\(545\) −10.0000 −0.428353
\(546\) 0 0
\(547\) −19.0000 −0.812381 −0.406191 − 0.913788i \(-0.633143\pi\)
−0.406191 + 0.913788i \(0.633143\pi\)
\(548\) 16.0000 0.683486
\(549\) −5.00000 −0.213395
\(550\) 0 0
\(551\) −8.00000 −0.340811
\(552\) 0 0
\(553\) 10.0000 0.425243
\(554\) 0 0
\(555\) 1.00000 0.0424476
\(556\) −24.0000 −1.01783
\(557\) −2.00000 −0.0847427 −0.0423714 − 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 0 0
\(559\) 12.0000 0.507546
\(560\) −4.00000 −0.169031
\(561\) 2.00000 0.0844401
\(562\) 0 0
\(563\) −17.0000 −0.716465 −0.358232 − 0.933632i \(-0.616620\pi\)
−0.358232 + 0.933632i \(0.616620\pi\)
\(564\) −18.0000 −0.757937
\(565\) −6.00000 −0.252422
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −39.0000 −1.63497 −0.817483 − 0.575953i \(-0.804631\pi\)
−0.817483 + 0.575953i \(0.804631\pi\)
\(570\) 0 0
\(571\) −38.0000 −1.59025 −0.795125 − 0.606445i \(-0.792595\pi\)
−0.795125 + 0.606445i \(0.792595\pi\)
\(572\) 12.0000 0.501745
\(573\) 27.0000 1.12794
\(574\) 0 0
\(575\) −3.00000 −0.125109
\(576\) −8.00000 −0.333333
\(577\) −14.0000 −0.582828 −0.291414 − 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) 0 0
\(579\) −23.0000 −0.955847
\(580\) −8.00000 −0.332182
\(581\) 5.00000 0.207435
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 3.00000 0.124035
\(586\) 0 0
\(587\) 20.0000 0.825488 0.412744 − 0.910847i \(-0.364570\pi\)
0.412744 + 0.910847i \(0.364570\pi\)
\(588\) 2.00000 0.0824786
\(589\) 10.0000 0.412043
\(590\) 0 0
\(591\) 18.0000 0.740421
\(592\) −4.00000 −0.164399
\(593\) −11.0000 −0.451716 −0.225858 − 0.974160i \(-0.572519\pi\)
−0.225858 + 0.974160i \(0.572519\pi\)
\(594\) 0 0
\(595\) −1.00000 −0.0409960
\(596\) 22.0000 0.901155
\(597\) 12.0000 0.491127
\(598\) 0 0
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 14.0000 0.571072 0.285536 − 0.958368i \(-0.407828\pi\)
0.285536 + 0.958368i \(0.407828\pi\)
\(602\) 0 0
\(603\) −12.0000 −0.488678
\(604\) −38.0000 −1.54620
\(605\) −7.00000 −0.284590
\(606\) 0 0
\(607\) 40.0000 1.62355 0.811775 − 0.583970i \(-0.198502\pi\)
0.811775 + 0.583970i \(0.198502\pi\)
\(608\) 0 0
\(609\) 4.00000 0.162088
\(610\) 0 0
\(611\) −27.0000 −1.09230
\(612\) −2.00000 −0.0808452
\(613\) 6.00000 0.242338 0.121169 − 0.992632i \(-0.461336\pi\)
0.121169 + 0.992632i \(0.461336\pi\)
\(614\) 0 0
\(615\) −11.0000 −0.443563
\(616\) 0 0
\(617\) 7.00000 0.281809 0.140905 − 0.990023i \(-0.454999\pi\)
0.140905 + 0.990023i \(0.454999\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 − 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 10.0000 0.401610
\(621\) 3.00000 0.120386
\(622\) 0 0
\(623\) 14.0000 0.560898
\(624\) −12.0000 −0.480384
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −4.00000 −0.159745
\(628\) 20.0000 0.798087
\(629\) −1.00000 −0.0398726
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 − 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 0 0
\(633\) 14.0000 0.556450
\(634\) 0 0
\(635\) −8.00000 −0.317470
\(636\) 0 0
\(637\) 3.00000 0.118864
\(638\) 0 0
\(639\) −2.00000 −0.0791188
\(640\) 0 0
\(641\) 10.0000 0.394976 0.197488 − 0.980305i \(-0.436722\pi\)
0.197488 + 0.980305i \(0.436722\pi\)
\(642\) 0 0
\(643\) 6.00000 0.236617 0.118308 − 0.992977i \(-0.462253\pi\)
0.118308 + 0.992977i \(0.462253\pi\)
\(644\) −6.00000 −0.236433
\(645\) −4.00000 −0.157500
\(646\) 0 0
\(647\) 12.0000 0.471769 0.235884 − 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 0 0
\(649\) 24.0000 0.942082
\(650\) 0 0
\(651\) −5.00000 −0.195965
\(652\) −38.0000 −1.48819
\(653\) −27.0000 −1.05659 −0.528296 − 0.849060i \(-0.677169\pi\)
−0.528296 + 0.849060i \(0.677169\pi\)
\(654\) 0 0
\(655\) −3.00000 −0.117220
\(656\) 44.0000 1.71791
\(657\) −4.00000 −0.156055
\(658\) 0 0
\(659\) −44.0000 −1.71400 −0.856998 − 0.515319i \(-0.827673\pi\)
−0.856998 + 0.515319i \(0.827673\pi\)
\(660\) −4.00000 −0.155700
\(661\) 14.0000 0.544537 0.272268 − 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 0 0
\(663\) −3.00000 −0.116510
\(664\) 0 0
\(665\) 2.00000 0.0775567
\(666\) 0 0
\(667\) −12.0000 −0.464642
\(668\) −4.00000 −0.154765
\(669\) 15.0000 0.579934
\(670\) 0 0
\(671\) 10.0000 0.386046
\(672\) 0 0
\(673\) 5.00000 0.192736 0.0963679 − 0.995346i \(-0.469277\pi\)
0.0963679 + 0.995346i \(0.469277\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 8.00000 0.307692
\(677\) 34.0000 1.30673 0.653363 − 0.757045i \(-0.273358\pi\)
0.653363 + 0.757045i \(0.273358\pi\)
\(678\) 0 0
\(679\) 2.00000 0.0767530
\(680\) 0 0
\(681\) −2.00000 −0.0766402
\(682\) 0 0
\(683\) 28.0000 1.07139 0.535695 − 0.844411i \(-0.320050\pi\)
0.535695 + 0.844411i \(0.320050\pi\)
\(684\) 4.00000 0.152944
\(685\) −8.00000 −0.305664
\(686\) 0 0
\(687\) −14.0000 −0.534133
\(688\) 16.0000 0.609994
\(689\) 0 0
\(690\) 0 0
\(691\) 15.0000 0.570627 0.285313 − 0.958434i \(-0.407902\pi\)
0.285313 + 0.958434i \(0.407902\pi\)
\(692\) 12.0000 0.456172
\(693\) 2.00000 0.0759737
\(694\) 0 0
\(695\) 12.0000 0.455186
\(696\) 0 0
\(697\) 11.0000 0.416655
\(698\) 0 0
\(699\) −3.00000 −0.113470
\(700\) 2.00000 0.0755929
\(701\) −41.0000 −1.54855 −0.774274 − 0.632850i \(-0.781885\pi\)
−0.774274 + 0.632850i \(0.781885\pi\)
\(702\) 0 0
\(703\) 2.00000 0.0754314
\(704\) 16.0000 0.603023
\(705\) 9.00000 0.338960
\(706\) 0 0
\(707\) 8.00000 0.300871
\(708\) −24.0000 −0.901975
\(709\) −12.0000 −0.450669 −0.225335 − 0.974281i \(-0.572348\pi\)
−0.225335 + 0.974281i \(0.572348\pi\)
\(710\) 0 0
\(711\) −10.0000 −0.375029
\(712\) 0 0
\(713\) 15.0000 0.561754
\(714\) 0 0
\(715\) −6.00000 −0.224387
\(716\) 38.0000 1.42013
\(717\) 15.0000 0.560185
\(718\) 0 0
\(719\) −45.0000 −1.67822 −0.839108 − 0.543964i \(-0.816923\pi\)
−0.839108 + 0.543964i \(0.816923\pi\)
\(720\) 4.00000 0.149071
\(721\) −16.0000 −0.595871
\(722\) 0 0
\(723\) −17.0000 −0.632237
\(724\) 2.00000 0.0743294
\(725\) 4.00000 0.148556
\(726\) 0 0
\(727\) −37.0000 −1.37225 −0.686127 − 0.727482i \(-0.740691\pi\)
−0.686127 + 0.727482i \(0.740691\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 4.00000 0.147945
\(732\) −10.0000 −0.369611
\(733\) 11.0000 0.406294 0.203147 − 0.979148i \(-0.434883\pi\)
0.203147 + 0.979148i \(0.434883\pi\)
\(734\) 0 0
\(735\) −1.00000 −0.0368856
\(736\) 0 0
\(737\) 24.0000 0.884051
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 − 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 2.00000 0.0735215
\(741\) 6.00000 0.220416
\(742\) 0 0
\(743\) −51.0000 −1.87101 −0.935504 − 0.353315i \(-0.885054\pi\)
−0.935504 + 0.353315i \(0.885054\pi\)
\(744\) 0 0
\(745\) −11.0000 −0.403009
\(746\) 0 0
\(747\) −5.00000 −0.182940
\(748\) 4.00000 0.146254
\(749\) −13.0000 −0.475010
\(750\) 0 0
\(751\) 50.0000 1.82453 0.912263 − 0.409605i \(-0.134333\pi\)
0.912263 + 0.409605i \(0.134333\pi\)
\(752\) −36.0000 −1.31278
\(753\) −16.0000 −0.583072
\(754\) 0 0
\(755\) 19.0000 0.691481
\(756\) −2.00000 −0.0727393
\(757\) −34.0000 −1.23575 −0.617876 − 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) 0 0
\(759\) −6.00000 −0.217786
\(760\) 0 0
\(761\) −6.00000 −0.217500 −0.108750 − 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 0 0
\(763\) 10.0000 0.362024
\(764\) 54.0000 1.95365
\(765\) 1.00000 0.0361551
\(766\) 0 0
\(767\) −36.0000 −1.29988
\(768\) −16.0000 −0.577350
\(769\) −16.0000 −0.576975 −0.288487 − 0.957484i \(-0.593152\pi\)
−0.288487 + 0.957484i \(0.593152\pi\)
\(770\) 0 0
\(771\) 7.00000 0.252099
\(772\) −46.0000 −1.65558
\(773\) 39.0000 1.40273 0.701366 − 0.712801i \(-0.252574\pi\)
0.701366 + 0.712801i \(0.252574\pi\)
\(774\) 0 0
\(775\) −5.00000 −0.179605
\(776\) 0 0
\(777\) −1.00000 −0.0358748
\(778\) 0 0
\(779\) −22.0000 −0.788232
\(780\) 6.00000 0.214834
\(781\) 4.00000 0.143131
\(782\) 0 0
\(783\) −4.00000 −0.142948
\(784\) 4.00000 0.142857
\(785\) −10.0000 −0.356915
\(786\) 0 0
\(787\) 44.0000 1.56843 0.784215 − 0.620489i \(-0.213066\pi\)
0.784215 + 0.620489i \(0.213066\pi\)
\(788\) 36.0000 1.28245
\(789\) −24.0000 −0.854423
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) −15.0000 −0.532666
\(794\) 0 0
\(795\) 0 0
\(796\) 24.0000 0.850657
\(797\) 27.0000 0.956389 0.478195 − 0.878254i \(-0.341291\pi\)
0.478195 + 0.878254i \(0.341291\pi\)
\(798\) 0 0
\(799\) −9.00000 −0.318397
\(800\) 0 0
\(801\) −14.0000 −0.494666
\(802\) 0 0
\(803\) 8.00000 0.282314
\(804\) −24.0000 −0.846415
\(805\) 3.00000 0.105736
\(806\) 0 0
\(807\) 21.0000 0.739235
\(808\) 0 0
\(809\) 2.00000 0.0703163 0.0351581 − 0.999382i \(-0.488807\pi\)
0.0351581 + 0.999382i \(0.488807\pi\)
\(810\) 0 0
\(811\) 31.0000 1.08856 0.544279 − 0.838905i \(-0.316803\pi\)
0.544279 + 0.838905i \(0.316803\pi\)
\(812\) 8.00000 0.280745
\(813\) −26.0000 −0.911860
\(814\) 0 0
\(815\) 19.0000 0.665541
\(816\) −4.00000 −0.140028
\(817\) −8.00000 −0.279885
\(818\) 0 0
\(819\) −3.00000 −0.104828
\(820\) −22.0000 −0.768273
\(821\) 22.0000 0.767805 0.383903 − 0.923374i \(-0.374580\pi\)
0.383903 + 0.923374i \(0.374580\pi\)
\(822\) 0 0
\(823\) 23.0000 0.801730 0.400865 − 0.916137i \(-0.368710\pi\)
0.400865 + 0.916137i \(0.368710\pi\)
\(824\) 0 0
\(825\) 2.00000 0.0696311
\(826\) 0 0
\(827\) 13.0000 0.452054 0.226027 − 0.974121i \(-0.427426\pi\)
0.226027 + 0.974121i \(0.427426\pi\)
\(828\) 6.00000 0.208514
\(829\) −26.0000 −0.903017 −0.451509 − 0.892267i \(-0.649114\pi\)
−0.451509 + 0.892267i \(0.649114\pi\)
\(830\) 0 0
\(831\) −7.00000 −0.242827
\(832\) −24.0000 −0.832050
\(833\) 1.00000 0.0346479
\(834\) 0 0
\(835\) 2.00000 0.0692129
\(836\) −8.00000 −0.276686
\(837\) 5.00000 0.172825
\(838\) 0 0
\(839\) 40.0000 1.38095 0.690477 − 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) 1.00000 0.0344418
\(844\) 28.0000 0.963800
\(845\) −4.00000 −0.137604
\(846\) 0 0
\(847\) 7.00000 0.240523
\(848\) 0 0
\(849\) −28.0000 −0.960958
\(850\) 0 0
\(851\) 3.00000 0.102839
\(852\) −4.00000 −0.137038
\(853\) −26.0000 −0.890223 −0.445112 − 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) 0 0
\(855\) −2.00000 −0.0683986
\(856\) 0 0
\(857\) 18.0000 0.614868 0.307434 − 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) 44.0000 1.50126 0.750630 − 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) −8.00000 −0.272798
\(861\) 11.0000 0.374879
\(862\) 0 0
\(863\) −14.0000 −0.476566 −0.238283 − 0.971196i \(-0.576585\pi\)
−0.238283 + 0.971196i \(0.576585\pi\)
\(864\) 0 0
\(865\) −6.00000 −0.204006
\(866\) 0 0
\(867\) −1.00000 −0.0339618
\(868\) −10.0000 −0.339422
\(869\) 20.0000 0.678454
\(870\) 0 0
\(871\) −36.0000 −1.21981
\(872\) 0 0
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) −8.00000 −0.270295
\(877\) −2.00000 −0.0675352 −0.0337676 − 0.999430i \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) 0 0
\(879\) −14.0000 −0.472208
\(880\) −8.00000 −0.269680
\(881\) −14.0000 −0.471672 −0.235836 − 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) 0 0
\(883\) −44.0000 −1.48072 −0.740359 − 0.672212i \(-0.765344\pi\)
−0.740359 + 0.672212i \(0.765344\pi\)
\(884\) −6.00000 −0.201802
\(885\) 12.0000 0.403376
\(886\) 0 0
\(887\) −42.0000 −1.41022 −0.705111 − 0.709097i \(-0.749103\pi\)
−0.705111 + 0.709097i \(0.749103\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) 30.0000 1.00447
\(893\) 18.0000 0.602347
\(894\) 0 0
\(895\) −19.0000 −0.635100
\(896\) 0 0
\(897\) 9.00000 0.300501
\(898\) 0 0
\(899\) −20.0000 −0.667037
\(900\) −2.00000 −0.0666667
\(901\) 0 0
\(902\) 0 0
\(903\) 4.00000 0.133112
\(904\) 0 0
\(905\) −1.00000 −0.0332411
\(906\) 0 0
\(907\) −40.0000 −1.32818 −0.664089 − 0.747653i \(-0.731180\pi\)
−0.664089 + 0.747653i \(0.731180\pi\)
\(908\) −4.00000 −0.132745
\(909\) −8.00000 −0.265343
\(910\) 0 0
\(911\) 4.00000 0.132526 0.0662630 − 0.997802i \(-0.478892\pi\)
0.0662630 + 0.997802i \(0.478892\pi\)
\(912\) 8.00000 0.264906
\(913\) 10.0000 0.330952
\(914\) 0 0
\(915\) 5.00000 0.165295
\(916\) −28.0000 −0.925146
\(917\) 3.00000 0.0990687
\(918\) 0 0
\(919\) 8.00000 0.263896 0.131948 − 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) 0 0
\(921\) −20.0000 −0.659022
\(922\) 0 0
\(923\) −6.00000 −0.197492
\(924\) 4.00000 0.131590
\(925\) −1.00000 −0.0328798
\(926\) 0 0
\(927\) 16.0000 0.525509
\(928\) 0 0
\(929\) −1.00000 −0.0328089 −0.0164045 − 0.999865i \(-0.505222\pi\)
−0.0164045 + 0.999865i \(0.505222\pi\)
\(930\) 0 0
\(931\) −2.00000 −0.0655474
\(932\) −6.00000 −0.196537
\(933\) 5.00000 0.163693
\(934\) 0 0
\(935\) −2.00000 −0.0654070
\(936\) 0 0
\(937\) −47.0000 −1.53542 −0.767712 − 0.640796i \(-0.778605\pi\)
−0.767712 + 0.640796i \(0.778605\pi\)
\(938\) 0 0
\(939\) 10.0000 0.326338
\(940\) 18.0000 0.587095
\(941\) 27.0000 0.880175 0.440087 − 0.897955i \(-0.354947\pi\)
0.440087 + 0.897955i \(0.354947\pi\)
\(942\) 0 0
\(943\) −33.0000 −1.07463
\(944\) −48.0000 −1.56227
\(945\) 1.00000 0.0325300
\(946\) 0 0
\(947\) 57.0000 1.85225 0.926126 − 0.377215i \(-0.123118\pi\)
0.926126 + 0.377215i \(0.123118\pi\)
\(948\) −20.0000 −0.649570
\(949\) −12.0000 −0.389536
\(950\) 0 0
\(951\) 5.00000 0.162136
\(952\) 0 0
\(953\) −50.0000 −1.61966 −0.809829 − 0.586665i \(-0.800440\pi\)
−0.809829 + 0.586665i \(0.800440\pi\)
\(954\) 0 0
\(955\) −27.0000 −0.873699
\(956\) 30.0000 0.970269
\(957\) 8.00000 0.258603
\(958\) 0 0
\(959\) 8.00000 0.258333
\(960\) 8.00000 0.258199
\(961\) −6.00000 −0.193548
\(962\) 0 0
\(963\) 13.0000 0.418919
\(964\) −34.0000 −1.09507
\(965\) 23.0000 0.740396
\(966\) 0 0
\(967\) 22.0000 0.707472 0.353736 − 0.935345i \(-0.384911\pi\)
0.353736 + 0.935345i \(0.384911\pi\)
\(968\) 0 0
\(969\) 2.00000 0.0642493
\(970\) 0 0
\(971\) −16.0000 −0.513464 −0.256732 − 0.966483i \(-0.582646\pi\)
−0.256732 + 0.966483i \(0.582646\pi\)
\(972\) 2.00000 0.0641500
\(973\) −12.0000 −0.384702
\(974\) 0 0
\(975\) −3.00000 −0.0960769
\(976\) −20.0000 −0.640184
\(977\) −18.0000 −0.575871 −0.287936 − 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 0 0
\(979\) 28.0000 0.894884
\(980\) −2.00000 −0.0638877
\(981\) −10.0000 −0.319275
\(982\) 0 0
\(983\) −34.0000 −1.08443 −0.542216 − 0.840239i \(-0.682414\pi\)
−0.542216 + 0.840239i \(0.682414\pi\)
\(984\) 0 0
\(985\) −18.0000 −0.573528
\(986\) 0 0
\(987\) −9.00000 −0.286473
\(988\) 12.0000 0.381771
\(989\) −12.0000 −0.381578
\(990\) 0 0
\(991\) −58.0000 −1.84243 −0.921215 − 0.389053i \(-0.872802\pi\)
−0.921215 + 0.389053i \(0.872802\pi\)
\(992\) 0 0
\(993\) 17.0000 0.539479
\(994\) 0 0
\(995\) −12.0000 −0.380426
\(996\) −10.0000 −0.316862
\(997\) 42.0000 1.33015 0.665077 − 0.746775i \(-0.268399\pi\)
0.665077 + 0.746775i \(0.268399\pi\)
\(998\) 0 0
\(999\) 1.00000 0.0316386
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1785.2.a.g.1.1 ✓ 1
3.2 odd 2 5355.2.a.k.1.1 1
5.4 even 2 8925.2.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1785.2.a.g.1.1 ✓ 1 1.1 even 1 trivial
5355.2.a.k.1.1 1 3.2 odd 2
8925.2.a.r.1.1 1 5.4 even 2